Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems |
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Authors: | X X Huang J C Yao |
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Affiliation: | 1. School of Economics and Business Administration, Chongqing University, Chongqing, 400030, China 2. Center for General Education, Kaohsiung Medical University, Kaohsiung, 807, Taiwan
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Abstract: | In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set-valued optimization problem (with or without cone constraints) in which the decision space is finite-dimensional. The characterizations are expressed in terms of the coercivity of some scalar set-valued maps and the well-posedness of the set-valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite-dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set-valued optimization problems and the well-posedness of the original problem. |
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